Lattice model of invasive dynamics of short- and long- rhizomes grasses

Дата канвертавання27.04.2016
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Lattice model of invasive dynamics of short- and long- rhizomes grasses

N.V. Mikhailova , N.E. Bogdanova

Institute of Mathematical Problems of Biology, Moscow region, Pushchino, Institutskaja st., 4, Russia,
Key words: plant population, lattice model, invasion, seed and vegetative dispersal


The evaluation of the rate and mechanism of plants invasion on bare territory is one of the basic problems of vegetation restoration. The fixedness of plants, their spatially explicit positions and interaction with the nearest neighbors are the main characteristics of a plant population life. A plant population represents a spatially distributed set of separate individuals differing by size and age. These individuals are able to disseminate their offsprings by many ways and with diverse spatial pattern. These circumstances make plant populations to be describing by complex spatial dynamics (Komarov et al., 2003).

We used computer modeling for analyzing the plants spatial pattern arising from the seed and vegetative propagation.

Three species of grasses with various types of territory occupation were chosen: Aegopodium podagraria L., which is a competitor, Stellaria holostea L., which is a reactive species and a stress-tolerant Asarum europaeum L. The model is based on the field experimental data collected in the “Bryanskii Les” State reserve in the centre of European Russia.

Our model is a dynamic finite automata model being discrete in time and space. For simplicity we considered the plant population development from the ontogenetic point of view. Individual plant development may be treated as a number of successive stages (Uranov, 1975). The stages are marked out on the basis of morphological indicators reflecting functional importance of plants at different stages. Thus population may be described as a set of individuals belonging to certain age stages with their corresponding distribution - age stages spectrum.

The main rules of the model are:

  • a plant population consists of population elements (sprouts) which originate either from seeds or from vegetative dissemination;

  • plants are assigned to the bounded cells of a two-dimensional integer lattice, with no more then one sprout per cell;

  • cell size is different for each species and depends on the distance of dauther sprouts spreading

  • time step is equal to one year.

  • plants change their age states, in the certain age plants spread its daughter sprouts on neighbor cells;

  • on reaching the generative age state the sprout is flowering and it dies next year doing the corresponding cell vacate;

  • seed rain is defined as the probability of random appearance of a seedling plant in an empty cell being in dependence on the distance from blooming plant (some special assumptions about other mechanisms of seeds transportation such as zoochoric transfer is also included into the model);

The chosen species differ in the geometrical properties of propagation of sprouts, rate of expanding and intensity of appearance of sprouts from mother plant. The Aegopodium podagraria population development is affected by light condition. The dynamic pattern of sprouts and their relative abilities were simulated in correspondence with experimental data (Table).

Table. Measured experimental data used in the models

Species, Conditions

Sprout life longevity, years

Vegetative dissemination age

Vegetative dissemination radius, m/year

Sprout blossom age

The seed germination number per sprout/year

Aegopodium podagraria in shadowed conditions






Aegopodium podagraria in light conditions






Asarum europaeum






Stellaria holostea






Results and discussion

We investigated different scenarios: 1) the invasion of different species separately on bare soil with or without shadowing for Aegopodium podagraria species; 2) the plant propagation dependence on presence or absence of zoochoric transfer 3) the role of plant community resistance at invasion; we define it as the heterogeneity of a territory for plants’ occupation treated as presence of busy cells, which prevent sprout propagation and seed germination. Last experiments are carried out with different probabilities of busy cells with random distribution using Monte-Carlo techniques.

In first scenario we found that Aegopodium podagraria has the fastest growth of the population at invasion on bare soil. At shadowing Aegopodium podagraria species do not produce seeds and the rate of invasion slows down. whereas invasion rate of Stellaria holostea does not depend on seed germination.

All species have dependence on presence or absence of zoochoric transfer mostly Asarum europaeum, its rate ov invasion increases ten times in comparison with two-three times for others. The Asarum europaeum population without seeds does not spread.

All species have their specific dynamics at third scenario. If the probability of busy cells presence is more then some critical probability being specifically for the species from number of all cells, the invasion stops. Without seeds the species have different critical probability number: 0.5 for Aegopodium podagraria, 0.1 for Asarum europaeum and 0.8 for Stellaria holostea. Again, with seed generation Aegopodium podagraria and Asarum europaeum have the same critical probability number but Stellaria holostea has not critical probability number at all.


Populations of Aegopodium podagraria have the fastest growth in presence the seed production, Stellaria holostea is the most adaptive species because of its vegetative overgrowth mechanism. The survival of Asarum Europaeum populations need the zoochoric transfer.


This work was supported by the grant 02-04-48965 of Russian Foundation of Basic Researches.


Komarov A.S., Palenova M.M., Smirnova O.V. The concept of discrete description of plant ontogenesis and cellular automata models of plant population. Ecological Modelling, 170, p. 427-439, 2003.

Uranov A.A. Age spectrum of the phytocoenopopulations as a function of time and energetic wave processes. Biological Sciences, Moskow, 2, p. 7-34, 1975.

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